We know that orthogonal polynomials on the solid cone and hyperboloid are eigenfunctions of a second order
linear differential operator with the eigenvalues depending only on the degree of the polynomials [1]}, [2]}.
In particular, this means that all polynomials of degree \(n\)  are eigenfunctions independent of the choice of bases.
In contrast, the orthogonal
polynomials on the solid paraboloid, as those on the surface of the paraboloid, do not posses this property.
For polynomials \({\mathbf {Q}}_{m,{\mathbf {k}}}^n\) , we can find a differential operator for which the eigenvalues depend
on \(n\)  and \(m\)  but not on \({\mathbf {k}}\) , as seen in the following analogue of Proposition REF , where
we assume \({\beta }=0\) . The latter assumption is consistent with \({\beta }=-\frac{1}{2}\)  in Proposition REF 
because \(W_{{\beta },{\gamma },\mu }\)  contains the factor \(t^{{\beta }+\mu -\frac{1}{2}}\)  when writing in the form (REF )
and, for \(\mu =0\) , \(\varpi _0\)  is the Chebyshev weight function on the unit ball, or the projection of the
surface measure of \({\mathbb {S}}^d\)  onto \({\mathbb {B}}^d\) .
